To solve an inequality with fractions, multiply every term by the LCM of the denominators, isolate x and write the interval. If you multiply or divide by a negative number, the direction flips. So x/3 − 1/2 > x/4 gives x > 6, which is the interval (6, +∞).
The method is the one for equations with fractions, with one more rule: in an equation the sign never changes, but in an inequality it can. If you need to solve two equations at once, see systems of equations with fractions.
| Operation on both sides | The direction | Example |
|---|---|---|
| Add or subtract any number | Stays the same | x − 6 > 0 becomes x > 6 |
| Multiply or divide by a positive number | Stays the same | x/5 ≤ 2 becomes x ≤ 10 |
| Multiply or divide by a negative number | Flips | −3x ≥ 12 becomes x ≤ −4 |
Only the last row changes the sign, which is why mistakes cluster there. With the LCM of the denominators, which is always positive, clearing the fractions never flips anything.
Worked example: x/3 − 1/2 > x/4
Find all the values of x that satisfy the inequality and write them as an interval.
- Find the LCM. The denominators are 3, 2 and 4, and their LCM is 12.
- Multiply every term by 12. Since 12 is positive, the sign stays as it is: 4x − 6 > 3x.
- Collect the x terms. Subtract 3x from both sides: x − 6 > 0.
- Isolate x. Add 6 and you get x > 6.
- Write the interval. It is (6, +∞), with a parenthesis at 6 because the inequality is strict.
Result: x > 6, as an interval (6, +∞).
Check: with x = 7, the left side is 7/3 − 1/2 = 11/6 = 22/12 and the right side is 7/4 = 21/12, so the inequality holds. With x = 6 both sides equal 3/2, which is why 6 is not included.
The endpoint is the value where the two sides are equal. Here it is 6, because 4x − 6 = 3x holds only when x = 6.
A negative coefficient flips the sign
When x has a negative coefficient, you have to divide by a negative number to get x alone. That is the only step that changes the direction, and it is the property stated in the English Wikipedia article “Inequality (mathematics)”, section “Multiplication and division”: if c is negative and a < b, then ac > bc (accessed October 17, 2026).
Take −x/5 < 2. Multiply by −5: the direction goes from < to > and you get x > −10, that is, (−10, +∞). The endpoint −10 is not included: −(−10)/5 equals 2, and 2 is not less than 2.
A test value inside the interval catches this mistake almost every time. It does not prove the whole interval, but if it fails, the result is definitely wrong. The guide to checking results collects other checks.
A double inequality is solved in one go
An inequality like −1 < (x − 2)/3 ≤ 5/6 has the expression in the middle and two bounds. You do the same operation on all three parts. Multiply by 3, which is positive: −3 < x − 2 ≤ 5/2. Add 2 to all three parts: −1 < x ≤ 9/2. The interval is (−1, 9/2], with a parenthesis at −1 and a bracket at 9/2 = 4.5.
If the multiplier were negative, you would have to flip both signs and also swap the bounds, so that the smaller number ends up on the left.
How to check the interval with test values
Pick one value inside and two outside, one on each side, and substitute them into the original inequality. For the double inequality, the results are as follows.
| x | (x − 2)/3 | Does −1 < … ≤ 5/6 hold? |
|---|---|---|
| −1 | −1 | No, the endpoint −1 is excluded |
| 0 | −2/3 | Yes |
| 9/2 | 5/6 | Yes, the endpoint 9/2 is included |
| 5 | 1 | No, it exceeds 5/6 |
The table confirms both boundaries: a value inside works, a value outside does not, and each endpoint is in or out depending on whether the sign is ≤ or <. On the number line the interval is drawn with an open circle at −1 and a closed one at 9/2.
Intervals are written like this: < and > take parentheses, ≤ and ≥ take brackets, and infinity always takes a parenthesis. For an LCM of more than two numbers, the least common multiple calculator gives it directly, and the least common denominator tool returns the fractions already scaled up.
Practice
- Solve x/4 − 1 ≤ x/6.
- Solve −3x/4 > 6.
- Solve (x + 1)/2 < (x − 1)/3.
- Solve 0 ≤ (2x + 1)/3 < 1.
Show the answers
- LCM 12: 3x − 12 ≤ 2x, so x ≤ 12. Interval (−∞, 12].
- Multiply by −4/3, a negative number, and the direction flips: x < −8. Interval (−∞, −8).
- LCM 6: 3x + 3 < 2x − 2, so x < −5. Interval (−∞, −5).
- Multiply by 3: 0 ≤ 2x + 1 < 3. Subtract 1: −1 ≤ 2x < 2. Divide by 2: −1/2 ≤ x < 1. Interval [−1/2, 1).
When x appears in a denominator, the approach is different; the first question below explains it. To keep going with fraction algebra, the advanced page collects the guides in this block, and on fraction practice you can practice calculating with fractions.